A curve \( C \) is defined by the parameric equations: \( \begin{array}{l}x=3+4 \operatorname{Cos} \theta \\ y=1+4 \operatorname{Sin} \theta\end{array} \) (i) Show that the equation of \( C \) represents a circl (ii) State the centre and radius of the circle.

To solve the problem, we will follow these steps: ### Step 1: Show that the equation of \( C \) represents a circle The parametric equations given are: \[ x = 3 + 4 \cos \theta \] \[ y = 1 + 4 \sin \theta \] We can eliminate the parameter \( \theta \) to find a relationship between \( x \) and \( y \). 1. Rearranging the equations for \( \cos \theta \) and \( \sin \theta \): \[ \cos \theta = \frac{x - 3}{4} \] \[ \sin \theta = \frac{y - 1}{4} \] 2. Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \): \[ \left(\frac{x - 3}{4}\right)^2 + \left(\frac{y - 1}{4}\right)^2 = 1 \] 3. Multiplying through by \( 16 \) to eliminate the denominators: \[ (x - 3)^2 + (y - 1)^2 = 16 \] This is the standard form of the equation of a circle, which confirms that the curve \( C \) represents a circle. ### Step 2: State the centre and radius of the circle From the equation \( (x - 3)^2 + (y - 1)^2 = 16 \): - The center of the circle is given by the point \( (h, k) \) where \( h = 3 \) and \( k = 1 \). - The radius \( r \) is the square root of the right-hand side, which is \( \sqrt{16} = 4 \). ### Final Answer (i) The equation of \( C \) represents a circle. (ii) The center of the circle is \( (3, 1) \) and the radius is \( 4 \).